new model uncertainty, state uncertainty, and state-space...

Model Uncertainty, State Uncertainty, andState-space Models

Yulei Luo, Jun Nie, Eric R. Young

Abstract State-space models have been increasingly used to study macroeconomicand financial problems. A state-space representation consists of two equations, ameasurement equation which links the observed variables to unobserved state vari-ables and a transition equation describing the dynamics of the state variables. In thispaper, we show that a classic linear-quadratic macroeconomic framework which in-corporates two new assumptions can be analytically solved and explicitly mappedto a state-space representation. The two assumptions we consider are the modeluncertainty due to concerns for model misspecification (robustness) and the stateuncertainty due to limited information constraints (rational inattention). We showthat the state-space representation of the observable and unobservable can be usedto quantify the key parameters on the degree of model uncertainty. We provide ex-amples on how this framework can be used to study a range of interesting questionsin macroeconomics and international economics.

1 Introduction

State-space models have been broadly applied to study macroeconomic and finan-cial problems. For example, they have been applied to model unobserved trends, tomodel transition from one economic structure to another, to forecasting models, tostudy wage-rate behaviors, to estimate expected inflation, and to model time-varyingmonetary reaction functions.

Yulei LuoThe University of Hong Kong e-mail: [email protected]

Jun NieFederal Reserve Bank of Kansas City, 1 memorial drive, Kansas City, MO, 64196 e-mail:[email protected]

Eric R. YoungUniversity of Virginia, Charlottesville, VA 22904 e-mail: [email protected]

1

2 Yulei Luo, Jun Nie, Eric R. Young

A state-space model typically consists of two equations, a measurement equa-tion which links the observed variables to unobserved state variables and a tran-sition equation which describes the dynamics of the state variables. The Kalmanfilter, which provides a recursive way to compute the estimator of the unobservedcomponent based on the observed variables, is a useful tool to analyze state-spacemodels.

In this paper, we show that a classic linear-quadratic-Gaussian (LQG) macroeco-nomic framework which incorporates two new assumptions can still be analyticallysolved and explicitly mapped to a state-space representation.1 The two assump-tions we consider are model uncertainty due to concerns for model misspecification(robustness) and state uncertainty due to limited information constraints (rationalinattention). We show that the state-space representation of the observable and un-observable can be used to quantify the key parameters by simulating the model. Weprovide examples on how this framework can be used to study a range of interestingquestions in macroeconomics and international economics.

The remainder of the paper is organized as follows. Section 2 presents the gen-eral framework. Section 3 shows how to introduce the model uncertainty and stateuncertainty to this framework. Section 4 provide several applications how to applythis framework to address a range of macroeconomic and international questions.In addition, it shows how this framework has a state-space representation. And thisstate-space representation can be used to quantify the key parameters in differentmodels. Section 5 concludes.

2 Linear-quadratic-Gaussian State-space Models

The linear-quadratic-Gaussian framework has been widely used in macroeconomics.This specification leads to the optimal linear regulator problem, for which the Bell-man equation can be solved easily using matrix algebra. The general setup is asfollows. The objective function has a quadratic form,

max{xt}

E0

[∞

∑t=0

β t f (xt)

](1)

and the maximization is subjected to a linear constraint

g(xt ,yt ,yt+1) = 0, for all t (2)

where g(·) is a linear function, xt is the vector of control variables and yt is thevector of state variables.

1 Note that here “linear” means that the state transition equation is linear, “quadratic” means thatthe objective function is quadratic, and “Gaussian” means that the exogenous innovation is Gaus-sian.

Model Uncertainty, State Uncertainty, and State-space Models 3

Example 1 (A small-open economy version of Hall’s permanent income model).Let xt = {ct ,bt+1}, yt = {bt ,yt}, f (xt) = − 12 (c− ct)

2, g(xt ,yt ,yt+1) = Rbt + yt −ct − bt+1, where c is the bliss point, ct is consumption, R is the exogenous andconstant gross world interest rate, bt is the amount of the risk-free foreign bondheld at the beginning of period t, and yt is net income in period t and is definedas output minus investment and government spending. Then this becomes a small-open economy version of Hall’s permanent income model in which a representativeagent chooses the consumption to maximize his utility subject to the exogenousendowments. As the representative agent can borrow from the rest of the worldat a risk-free interest rate, the resource constraint need not bind every period. Ifwe remove this assumption, the model goes back to the permanent income modelstudied in Hall (1978).2

Example 2 (Barro’s tax-smoothing model). Barro (1979) proposed a simplerational expectations (RE) tax-smoothing model with only noncontingent debt inwhich the government spreads the burden of raising distortionary income taxesover time in order to minimize their welfare losses to address these questions.3

This tax-smoothing hypothesis has been widely used (to address various fiscalpolicies) and tested. The model also falls well into this linear-quadratic frame-work.4 Specifically, let xt = {τt ,Bt+1}, yt = {Yt ,Gt}, f (xt) =− 12 τ

2t , g(xt ,yt ,yt+1) =

RBt +Gt − τtYt −Bt+1, where E0 [·] is the government’s expectation conditional onits available and processed information set at time 0, β is the government’s sub-jective discount factor, τt is the tax rate, Bt is the amount of government debt, Gtis government spending, Yt is real GDP, and R is the gross interest rate. Here weassume that the welfare costs of taxation are proportional to the square of the taxrate.5

In general, the number of the state variables in these models can be more thanone. But in order to facilitate the introduction of robustness we reduce the abovemultivariate model with a general exogeneous process to a univariate model withiid innovations that can be solved in closed-form. Specifically, following Luo andYoung (2010) and Luo, Nie, and Young (2011a), we rewrite the model described by(1) and (2) as

max{zt ,st+1}∞t=0

{E0

[∞

∑t=0

β t f (zt)

]}(3)

subject tost+1 = Rst − zt +ζt+1, (4)

2 We take a small-open economy version of Hall’s model as we’ll use it to address some small-openeconomy issues in later sectors.3 It is worth noting that the tax-smoothing hypothesis (TSH) model is an analogy with the perma-nent income hypothesis (PIH) model in which consumers smooth consumption over time; tax ratesrespond to permanent changes in the public budgetary burden rather than transitory ones.4 For example, see Huang and Lin (1993), Ghosh (1995), and Cashin et al (2001).5 Following Barro (1979), Sargent (1987), Bohn (1989), and Huang and Lin (1993), we only needto impose the restriction, f ′ (τ)> 0 and f ′′ (τ)> 0, on the loss function, f (τ).


where both zt and st are single variables, and ζt+1 is the Gaussian innovation to thestate transition equation with mean 0 and variance ω2ζ .

For instance, for Example 1, the mapping is

zt = ct ,

st = bt +1R

∞

∑j=0

R− jEt [yt+ j] ,

ζt+1 =1R

∞

∑j=t+1

(1R

) j−(t+1)(Et+1−Et) [y j] .

And for Example 2, the mapping is

zt = τt ,

st = Et

[bt +

1

(1+n) R̃

∞

∑j=0

(1

R̃

) jgt+ j

],

ζt+1 =∞

∑j=0

(1

R̃

) j+1(Et+1−Et)

[gt+1+ j

],

where R̃ = R/(1+n) is the effective interest rate faced by the government, n is theGDP growth rate, bt and gt are government debt and government spending as a ratioof GDP.6

Finally, the recursive representation of the above problem is as follows.

v(st) = maxzt{ f (zt)+βEt [v(st+1)]} (5)

subject to:st+1 = Rst − zt +ζt+1, (6)

given s0.

3 Incorporating Model Uncertainty and State Uncertainty

In this section we show how to incorporate model uncertainty and state uncertaintyinto the framework presented in the previous section.

6 n is assumed to be constant.


3.1 Introducing Model Uncertainty

We focus on the model uncertainty due to a concern for model misspecification(robustness). Hansen and Sargent (1995, 2007a) first introduce robustness (a con-cern for model misspecification) into economic models. In robust control problems,agents are concerned about the possibility that their model is misspecified in a man-ner that is difficult to detect statistically; consequently, they choose their decisionsas if the subjective distribution over shocks was chosen by a malevolent nature inorder to minimize their expected utility (that is, the solution to a robust decision-maker’s problem is the equilibrium of a max-min game between the decision-makerand nature). Specifically, a robustness version of the model represented by (5) and(6) are

v(st) = maxzt

minνt

{f (zt)+β

[ϑν2t +Et [v(st+1)]

]}(7)

subject to the distorted transition equation (i.e., the worst-case model):

st+1 = Rst − zt +ζt+1 +ωζ νt , (8)

where νt distorts the mean of the innovation and ϑ > 0 controls how bad the errorcan be.7

3.2 Introducing State Uncertainty

In this section we introduce state uncertainty into the model we see in the previoussection. It will be seen that state uncertainty will further amplify the effect due tomodel uncertainty.8 We consider the model with imperfect state observation (stateuncertainty) due to finite information-processing capacity (rational inattention orRI). Sims (2003) first introduced RI into economics and argued that it is a plausiblemethod for introducing sluggishness, randomness, and delay into economic models.In his formulation agents have finite Shannon channel capacity, limiting their abilityto process signals about the true state of the world. As a result, an impulse to theeconomy induces only gradual responses by individuals, as their limited capacityrequires many periods to discover just how much the state has moved.

Under RI, consumers in the economy face both the usual flow budget constraintand information-processing constraint due to finite Shannon capacity first intro-duced by Sims (2003). As argued by Sims (2003, 2006), individuals with finite

7 Formally, this setup is a game between the decision-maker and a malevolent nature that choosesthe distortion process νt . ϑ ≥ 0 is a penalty parameter that restricts attention to a limited class ofdistortion processes; it can be mapped into an entropy condition that implies agents choose rulesthat are robust against processes which are close to the trusted one. In a later section we will applyan error detection approach to calibrate ϑ .8 This will be clearer when we go to the applications in later sections.


channel capacity cannot observe the state variables perfectly; consequently, theyreact to exogenous shocks incompletely and gradually. They need to choose theposterior distribution of the true state after observing the corresponding signal. Thischoice is in addition to the usual consumption choice that agents make in their utilitymaximization problem.9

Following Sims (2003), the consumer’s information-processing constraint can becharacterized by the following inequality:

H (st+1|It)−H (st+1|It+1)≤ κ, (9)

where κ is the consumer’s channel capacity, H (st+1|It) denotes the entropy of thestate prior to observing the new signal at t + 1, and H (st+1|It+1) is the entropyafter observing the new signal.10 The concept of entropy is from information theory,and it characterizes the uncertainty in a random variable. The right-hand side of (9),being the reduction in entropy, measures the amount of information in the new signalreceived at t +1. Hence, as a whole, (9) means that the reduction in the uncertaintyabout the state variable gained from observing a new signal is bounded from aboveby κ . Since the ex post distribution of st is a normal distribution, N

(ŝt ,σ2t

), (9) can

be reduced tolog |ψ2t |− log |σ2t+1| ≤ 2κ (10)

where ŝt is the conditional mean of the true state, and σ2t+1 = var [st+1|It+1] andψ2t = var [st+1|It ] are the posterior variance and prior variance of the state variable,respectively. To obtain (10), we use the fact that the entropy of a Gaussian randomvariable is equal to half of its logarithm variance plus a constant term.

It is straightforward to show that in the univariate case (10) has a unique steadystate σ2.11 In that steady state the consumer behaves as if observing a noisy mea-surement which is s∗t+1 = st+1 + ξt+1, where ξt+1 is the endogenous noise and itsvariance α2t = var [ξt+1|It ] is determined by the usual updating formula of the vari-ance of a Gaussian distribution based on a linear observation:

σ2t+1 = ψ2t −ψ2t

(ψ2t +α

2t)−1 ψ2t . (11)

Note that in the steady state σ2 = ψ2−ψ2(ψ2 +α2

)−1 ψ2, which can be solvedas α2 =

[(σ2)−1− (ψ2)−1]−1. Note that (11) implies that in the steady state σ2 =

ω2ζexp(2κ)−R2 and α

2 = var [ξt+1] =[ω2ζ+R

2σ2]σ2

ω2ζ+(R2−1)σ2

.

9 More generally, agents choose the joint distribution of consumption and current permanent in-come subject to restrictions about the transition from prior (the distribution before the currentsignal) to posterior (the distribution after the current signal). The budget constraint implies a linkbetween the distribution of consumption and the distribution of next period permanent income.10 We regard κ as a technological parameter. If the base for logarithms is 2, the unit used tomeasure information flow is a ‘bit’, and for the natural logarithm e the unit is a ‘nat’. 1 nat is equalto log2 e≈ 1.433 bits.11 Convergence requires that κ > log(R)≈ R−1; see Luo and Young (2010) for a discussion.


We now incorporate state uncertainty due to RI into the RB model proposed inthe last section. There two different ways to do it. The simpler way is to assume thatthe consumer only has doubts about the process for the shock to permanent incomeζt+1, but trusts his regular Kalman filter hitting the endogenous noise (ξt+1) andupdating the estimated state. In the next subsection, we will relax the assumptionthat the consumer trusts the Kalman filter equation which generates an additionaldimension along which the agents in the economy desire robustness.

The RB-RI model is formulated as

v̂(ŝt) = maxzt

minνt

{f (zt)+βEt

[ϑν2t + v̂(ŝt+1)

]}, (12)

subject to the (budget) constraint

st+1 = Rst − zt +ωζ νt +ζt+1 (13)

and the regular Kalman filter equation

ŝt+1 = (1−θ)(Rŝt − zt +ωζ νt

)+θ (st+1 +ξt+1) (14)

Notice that f (zt) is a quadratic function, so the model is in a linear-quadraticform. As to be shown in the next section, we can explicitly solve the optimal choicefor control variable zt and the worst case shock νt . After substituting these twosolutions into the transition equations for st and ŝt , it can easily be shown that themodel has a state-space representation.

3.2.1 Robust filtering under RI

It is clear that the Kalman filter under RI, (13), is not only affected by the funda-mental shock (ζt+1), but also affected by the endogenous noise (ξt+1) induced byfinite capacity; these noise shocks could be another source of the demand for robust-ness. We therefore need to consider this demand for robustness in the RB-RI model.By adding the additional concern for robustness developed here, we are able tostrengthen the effects of robustness on decisions.12 Specifically, we assume that theagent thinks that (14) is the approximating model. Following Hansen and Sargent(2007), we surround (14) with a set of alternative models to represent a preferencefor robustness:

ŝt+1 = Rŝt − zt +ωη νt +ηt+1. (15)

whereηt+1 = ϑR(st − ŝt)+ϑ(ζt+1 +ξt+1) (16)

and Et [ηt+1] = 0 because the expectation is conditional on the perceived signals andinattentive agents cannot perceive the lagged shocks perfectly.

12 Luo, Nie, and Young (2011a) use this approach to study the joint dynamics of consumption,income, and the current account.


Under RI the innovation ηt+1, (16), that the agent distrusts is composed of twoMA(∞) processes and includes the entire history of the exogenous income shockand the endogenous noise, {ζt+1,ζt , · · ·,ζ0;ξt+1,ξt , · · ·,ξ0}. The difference between(13)) and (15) is the third term; in (13) the coefficient on νt is ωζ while in (15) thecoefficient is ωη ; note that with θ < 1 and R > 1 it holds that ωζ < ωη .

The optimizing problem for this RB-RI model can be formulated as follows:

v̂(ŝt) = maxct

minνt

{f (zt)+βEt

[ϑν2t + v̂(ŝt+1)

]}(17)

subject to (15). (17) is a standard dynamic programming problem and can be easilysolved using the standard procedure.

4 Applications

This section provides several applications of the framework developed in Section3.13 In each application, the model can be mapped into the general framework pre-sented in the previous section. Using these examples, we show how this frameworkcan be analytically solved and can be explicitly mapped to a state-space representa-tion (Section 4.1). We also show that this state-space representation plays an impor-tant role in quantifying the model uncertainty and state uncertainty (Section 4.4).These applications show how model uncertainty (RB) and state uncertainty (RI orimperfect information) alter the results from the standard framework presented inSection 2.

4.1 Explaining Current Account Dynamics

Return in to Example 1 in Section 2. The model is a small-open economy versionof the permanent income model. The standard model is represented by (5) and (6),while the model incorporating model uncertainty and state uncertainty is representedby (12)-(14). (Notice that zt = ct and f (xt) =− 12 (c− ct)

2.)As shown in Luo et al (2011a), given ϑ and θ , the consumption function under

RB and RI isct =

R−11−Σ

ŝt −Σc

1−Σ, (18)

the mean of the worst-case shock is

ωη νt =(R−1)Σ

1−Σŝt −

Σ1−Σ

c, (19)

13 These illustrations are based on the research by Luo and Young (2010) and Luo, Nie and Young(2011a, 2011b, 2011c).


where ρs = 1−RΣ1−Σ ∈ (0,1), Σ = Rω2η/(2ϑ), ω2η = var [ηt+1] = θ1−(1−θ)R2 ω

2ζ .

Substituting (19) into (13) and combining with (14), the observed st and unob-served ŝt are governed by the following two equations

st − ŝt =(1−θ)ζt

1− (1−θ)R ·L− θξt

1− (1−θ)R ·L(20)

ŝt+1 = ρsŝt +ηt+1. (21)

whereηt+1 = θR(st − ŝt)+θ (ζt+1 +ξt+1) (22)

Thus, it’s clear to see that (20) and (21) form a state-space representation themodel in which (20) is the measurement equation that links the observed variablest to unobserved variable ŝt and (21) is the transition equation which describes thedynamics of ŝt .

Notice that Σ measures the effects of both model uncertainty and state uncer-tainty, which is bounded by 0 and 1.14 As argued in Sims (2003), although therandomness in an individual’s response to aggregate shocks will be idiosyncraticbecause it arises from the individual’s own information-processing constraint, thereis likely a significant common component. The intuition is that people’s needs forcoding macroeconomic information efficiently are similar, so they rely on commonsources of coded information. Therefore, the common term of the idiosyncratic er-ror, ξ t , lies between 0 and the part of the idiosyncratic error, ξt , caused by thecommon shock to permanent income, ζt . Formally, assume that ξt consists of twoindependent noises: ξt = ξ t + ξ it , where ξ t = E i [ξt ] and ξ it are the common andidiosyncratic components of the error generated by ζt , respectively. A single param-eter,

λ =var[ξ t]

var [ξt ]∈ [0,1],

can be used to measure the common source of coded information on the aggregatecomponent (or the relative importance of ξ t vs. ξt ).15

Next, we briefly list the facts we focus on (Table 1). First, the correlation betweenthe current account and net income is positive but small (and insignificant whendetrended with the HP filter). Second, the relative volatility of the current account tonet income is smaller in emerging countries than in developed economies, althoughthe difference is not statistically significant when the series are detrended with theHP filter. Third, the persistence of the current account is smaller than that of netincome, and less persistent in emerging economies. And fourth, the volatility ofconsumption growth relative to income growth is larger in emerging economies thanin developed economies.

14 See Luo, Nie, and Young (2011a) for the proof.15 It is worth noting that the special case that λ = 1 can be viewed as a representative-agent modelin which we do not need to discuss the aggregation issue.


Finally, let’s compare the model implications, as summarized in Table 2. First,we have seen that in this case (λ = 1 and θ = 50%) the interaction of RB and RImake the model fit the data quite well along dimensions (3) and (4), while also quan-titatively improving the model’s predictions along dimensions (1) and (2). Second,this improvement does not preclude the model from matching the first two dimen-sions as well (i.e., the contemporaneous correlation between the current account andnet income and the volatility of the current account). For example, holding λ equalto 1 and further reducing θ can generate a smaller contemporaneous correlationbetween the current account and net income which is closer to the data. And hold-ing θ = 50% and reducing λ to 0.1 can make the relative volatility of the currentaccount to net income very close to the data.

4.2 Resolving The International Consumption Puzzle

The same framework can be used to address an old puzzle in the international eco-nomics literature. That is, the cross-country consumption correlations are very lowin the data (lower than the cross-country correlations of outputs) while standardmodels imply the opposite.16

To show the flexibility of the general framework summarized by (5) and (6), weslightly deviate from the assumption we used in the previous subsection (example 1)to introduce state uncertainty (SU). We assume that consumers in the model econ-omy cannot observe the true state st perfectly and only observes the noisy signal

s∗t = st +ξt , (23)

when making decisions, where ξt is the iid Gaussian noise due to imperfect obser-vations. The specification in (23) is standard in the signal extraction literature andcaptures the situation where agents happen or choose to have imperfect knowledgeof the underlying shocks.17 Since imperfect observations on the state lead to wel-fare losses, agents use the processed information to estimate the true state.18 Specif-ically, we assume that households use the Kalman filter to update the perceived stateŝt = Et [st ] after observing new signals in the steady state:

ŝt+1 = (1−θ)(Rŝt − ct)+θ (st+1 +ξt+1) , (24)

16 For example, Backus, Kehoe, and Kydland (1992) solve a two-country real business cyclesmodel and argue that the puzzle that empirical consumption correlations are actually lower thanoutput correlations is the most striking discrepancy between theory and data.17 For example, Muth (1960), Lucas (1972), Morris and Shin (2002), and Angeletos and La’O(2009). It is worth noting that this assumption is also consistent with the rational inattention ideathat ordinary people only devote finite information-processing capacity to processing financialinformation and thus cannot observe the states perfectly.18 See Luo (2008) for details about the welfare losses due to information imperfections within thepartial equilibrium permanent income hypothesis framework.


where θ is the Kalman gain (i.e., the observation weight).19In the signal extraction problem, the Kalman gain can be written as

θ =ϒΛ−1, (25)

where ϒ is the steady state value of the conditional variance of st+1, vart+1 [st+1],and is the variance of the noise, Λ = vart [ξt+1]. ϒ and Λ are linked by the followingequation which updates the conditional variance in the steady state:

Λ−1 =ϒ−1−Ψ−1, (26)

where Ψ is the steady state value of the ex ante conditional variance of st+1, Ψt =var t [st+1].

Multiplying ω2ζ on both sides of (26) and using the fact that Ψ = R2ϒ +ω2ζ , we

have

ω2ζ Λ−1 = ω2ζϒ

−1−[

R2(

ω2ζϒ−1)−1

+1]−1

, (27)

where ω2ζϒ−1 =

(ω2ζ Λ

−1)(

Λϒ−1).

Define SNR as π = ω2ζ Λ−1. We obtain the following equality linking SNR (π)

and the Kalman gain (θ):

π = θ(

11−θ

−R2). (28)

Solving for θ from the above equation yields

θ =−(1+π)+

√(1+π)2 +4R2 (π +R2)

2R2, (29)

where we omit the negative values of θ because both ϒ and Λ must be positive.Note that given π , we can pin down Λ using π = ω2ζ Λ

−1 and ϒ using (25) and (29).Combining (4) with (24), we obtain the following equation governing the per-

ceived state ŝt :ŝt+1 = Rŝt − ct +ηt+1, (30)

whereηt+1 = θR(st − ŝt)+θ (ζt+1 +ξt+1) (31)

is the innovation to the mean of the distribution of perceived permanent income,

st − ŝt =(1−θ)ζt

1− (1−θ)R ·L− θξt

1− (1−θ)R ·L(32)

19 Note that θ measures how much uncertainty about the state can be removed upon receiving thenew signals about the state.


is the estimation error where L is the lag operator, and Et [ηt+1] = 0.Note that ηt+1can be rewritten as

ηt+1 = θ[(

ζt+11− (1−θ)R ·L

)+

(ξt+1−

θRξt1− (1−θ)R ·L

)], (33)

where ω2ξ = var [ξt+1] =1θ

11/(1−θ)−R2 ω

2ζ . Expression (33) clearly shows that the es-

timation error reacts to the fundamental shock positively, while it reacts to the noiseshock negatively. In addition, the importance of the estimation error is decreasingwith θ . More specifically, as θ increases, the first term in (33) becomes less im-portant because (1−θ)ζt in the numerator decreases, and the second term alsobecomes less important because the importance of ξt decreases as θ increases.20

Although the assumption we use to introduce state uncertainty is different, thegeneral framework is still the same. More importantly, the solution strategy is alsothe same. Basically, we can explicitly derive the expressions for consumption andthe worst-case shock and then substitute them into (30). Together with (32), it formsa state-space representation of the model.

Table 3 reports the implied consumption correlations (between the domesticcountry and ROW) between the RE, RB, and RB-SU models. There are two interest-ing observations in the table. First, given the degrees of RB and SU (θ), corr(ct ,c∗t )decreases with the aggregation factor (λ ). Second, when λ is positive (even if itis very small, e.g., 0.1 in the table), corr(ct ,c∗t ) is decreasing with the degree ofinattention (i.e., increasing with θ ). The intuition is that when there are commonnoises, the effect of the noises could dominate the effect of gradual consumptionadjustments on cross-country consumption correlations.

As we can see from Table 3, for all the countries we consider here, introduc-ing SU into the RB model can make the model better fit the data on consump-tion correlations at many combinations of the parameter values. For example, forItaly, when θ = 60% (60% of the uncertainty is removed upon receiving a newsignal about the innovation to permanent income) and λ = 1, the RB-SU modelpredicts that corr(ct ,c∗t ) = 0.27, which is very close to the empirical counterpart,0.25.21 For France, when θ = 90% and λ = 0.5, the RB-SU model predicts thatcorr(ct ,c∗t ) = 0.46, which exactly matches the empirical counterpart. Note that asmall value of θ can be rationalized by examining the welfare effects of finite chan-nel capacity.22

20 Note that when θ = 1, var [ξt+1] = 0.21 For example, Adam (2005) found θ = 40% based on the response of aggregate output to mon-etary policy shocks. Luo (2008) found that if θ = 50%, the otherwise standard permanent incomemodel can generate realistic relative volatility of consumption to labor income.22 See Luo and Young (2010) for details about the welfare losses due to imperfect observations inthe RB model; they are uniformly small.


4.3 Other Possible Applications

This linear-quadratic framework which incorporates model uncertainty (due to RB)and state uncertainty (either due to RI or imperfect information) can be applied tostudy other topics as well. We will briefly discuss several more in this subsection.We will not write down the model equations again as we have shown in Section 2and 3 that these models can be written in a similar framework.

First, as shown in the previous section, model uncertainty due to RB is partic-ularly promising and interesting for studying emerging and developed small-openeconomies because it has the potential to generate the different joint behaviors ofconsumption and current accounts observed across the two groups of economies.This novel theoretical contribution can also be used to address the observed U.S.Great Moderation in which the volatility of output changed after 1984. Specifically,this feature can be used to address different macroeconomic dynamics (e.g., con-sumption volatility) given that output volatility changed before and after the GreatModeration.

Second, inventories in the standard production smoothing model can be viewedas a stabilizing factor. Cost-mininizing firms facing sales fluctuations smooth pro-duction by adjusting their inventories. As a result, production is less volatile thansales. However, in the data, real GDP is more volatile than final sales measuredby real GDP minus inventory investment. The existing studies find supportive evi-dence that real GNP is more volatile than final sales in industry-level data. The keyquestion is that if cost-minimizing firms use inventories to smooth their production,why is production more volatile than sales? In the future research, we can examinewhether introducing RB can help improve the prediction of an otherwise standardproduction smoothing model with inventories on the joint dynamics of inventories,production, and sales.

Third, as shown in Luo, Nie, and Young (2011c), the standard tax-smoothingmodel proposed by Barro (1979) cannot explain the observed volatility of the taxrates and the joint behavior of the government spending and deficits. As shown inExample 2 of Section (2), the tax-smoothing model used in the literature falls wellinto the linear-quadratic framework we described. It’s easy to show that the samemechanisms presented in Section 4.1 and 4.2 will work in the tax-smoothing modelwhich incorporates model uncertainty and state uncertainty. Specifically, Luo, Nie,and Young (2011c) shows that it can help the standard model to better explain therelative volatility of the changes in tax rates to government spending and the co-movement between government deficits and spending in the data.

Fourth, this framework can also be extended to study optimal monetary policyunder model uncertainty and imperfect state observation. A central bank sets nomi-nal interest rate to minimize prices fluctuations and the output gap (i.e., the deviationof the output from the potential maximum output level). Following the literature, thestandard objective function of a central bank can be described by a quadratic func-tion which is a weighted average of the deviation of the inflation from its target and


the output gap.23 Therefore, the framework presented in this paper can be used tostudy optimal monetary policy when a central bank has concerns that the model ismisspecified and it faces noisy data when making decisions.24

4.4 Quantifying Model Uncertainty

One remaining question from previous sections is how to quantify the incorporateddegree of model uncertainty.25 In this section, we will show how to use the state-space representation of st and ŝt to simulate the model and calibrate the key param-eters. For convenience and consistence, we continue to use the small-open economymodel described in Example 1 as the illustration example.

Let model A denote the approximating model and model B be the distortedmodel. Define pA as

pA = Prob(

log(

LALB

)< 0∣∣∣∣A) , (34)

where log(

LALB

)is the log-likelihood ratio. When model A generates the data, pA

measures the probability that a likelihood ratio test selects model B. In this case,we call pA the probability of the model detection error. Similarly, when model Bgenerates the data, we can define pB as

pB = Prob(

log(

LALB

)> 0∣∣∣∣B) . (35)

Following Hansen, Sargent, and Wang (2002) and Hansen and Sargent (2007b),the detection error probability, p, is defined as the average of pA and pB:

p(ϑ) =12(pA + pB) , (36)

where ϑ is the robustness parameter used to generate model B. Given this definition,we can see that 1− p measures the probability that econometricians can distinguishthe approximating model from the distorted model.

Now we show how to compute the model detection error probability due to modeluncertainty and state uncertainty.

23 For example, see Svensson (2000), Gali and Monacelli (2005), Walsh (2005), Leitemo andSoderstrom (2008a,b).24 For the examples of the model equations describing the inflation and output dynamics in a closedeconomy, see Leitemo and Soderstrom (2008a).25 This includes the two versions of the model presented in previous sections which incorporatesthe model uncertainty due to RB: one uses the regular Kalman filter; the other one assumes thatthe agent does not trust the Kalman filter either (robust filtering).


In the model with both the RB preference and RI, the approximating model canbe written as

st+1 = Rst − ct +ζt+1, (37)ŝt+1 = (1−θ)(Rŝt − ct)+θ (st+1 +λξt+1) , (38)

and the distorted model is

st+1 = Rst − ct +ζt+1 +ωζ νt , (39)ŝt+1 = (1−θ)

(Rŝt − ct +ωζ νt

)+θ (st+1 +λξt+1) , (40)

where we remind the reader that λ = var[ξ t ]var[ξt ] ∈ [0,1] is the parameter measuring therelative importance of ξ t vs. ξt .

After substituting the consumption function and the worst-case shock expressioninto (38) and (40) we can put the equations in the following matrix form:[

st+1ŝt+1

]=

[R − R−11−Σ

θR 1−R+R(1−θ)(1−Σ)1−Σ

][stŝt

]+

[ζt+1

θ (ζt+1 +λξt+1)

]+

[ Σ1−Σ c

Σ1−Σ c

](41)

and [st+1ŝt+1

]=

[R −(R−1)

θR 1−θR

][stŝt

]+

[ζt+1

θ (ζt+1 +λξt+1)

]. (42)

Given the RB parameter, ϑ , and RI parameter, θ , we can compute pA and pB andthus the detection error probability as follows.

1. Simulate {st}Tt=0 using (41) and (42) a large number of times. The number ofperiods used in the simulation, T , is set to be the actual length of the data foreach individual country.

2. Count the number of times that log(

LALB

)< 0∣∣∣A and log(LALB)> 0∣∣∣B are each

satisfied.3. Determine pA and pB as the fractions of realizations for which log

(LALB

)< 0∣∣∣A

and log(

LALB

)> 0∣∣∣B, respectively.

4.5 Discussions: Risk-sensitivity and Robustness under RationalInattention

Risk-sensitivity (RS) was first introduced into the LQG framework by Jacobson(1973) and extended by Whittle (1981, 1990). Exploiting the recursive utility frame-work of Epstein and Zin (1989), Hansen and Sargent (1995) introduce discount-ing into the RS specification and show that the resulting decision rules are time-invariant. In the RS model agents effectively compute expectations through a dis-


torted lens, increasing their effective risk aversion by overweighting negative out-comes. The resulting decision rules depend explicitly on the variance of the shocks,producing precautionary savings, but the value functions are still quadratic func-tions of the states.26 In HST (1999) and Hansen and Sargent (2007), they interpretthe RS preference in terms of a concern about model uncertainty (robustness or RB)and argue that RS introduces precautionary savings because RS consumers want toprotect themselves against model specification errors.

Following Luo and Young (2010), we formulate an RI version of risk-sensitivecontrol based on recursive preferences with an exponential certainty equivalencefunction as follows:

v̂(ŝt) = maxct

{−1

2(ct − c)2 +βRt [v̂(ŝt+1)]

}(43)

subject to the budget constraint (6) and the Kalman filter equation 14. The distortedexpectation operator is now given by

Rt [v̂(ŝt+1)] =−1α

logEt [exp(−α v̂(ŝt+1))] ,

where s0| I 0 ∼ N(ŝ0,σ2

), ŝt = Et [st ] is the perceived state variable, θ is the opti-

mal weight on the new observation of the state, and ξt+1 is the endogenous noise.The optimal choice of the weight θ is given by θ (κ) = 1−1/exp(2κ) ∈ [0,1]. Thefollowing proposition summarizes the solution to the RI-RS model when βR = 1:

Proposition 1. Given finite channel capacity κ and the degree of risk-sensitivity α ,the consumption function of a risk-sensitive consumer under RI

ct =R−11−Π

ŝt −Πc

1−Π, (44)

where

Π = Rαω2η ∈ (0,1) , (45)

ω2η = var [ηt+1] =θ

1− (1−θ)R2ω2ζ , (46)

ηt+1 is defined in (16), and θ (κ) = 1−1/exp(2κ).

Comparing (18) and (44), it is straightforward to show that it is impossible to dis-tinguish between RB and RS under RI using only consumption-savings decisions.

Proposition 2. Let the following expression hold:

α =1

2ϑ. (47)

26 Formally, one can view risk-sensitive agents as ones who have non-state-separable preferences,as in Epstein and Zin (1989), but with a value for the intertemporal elasticity of substitution equalto one.


Then consumption and savings are identical in the RS-RI and RB-RI models.

Note that (47) is exactly the same as the observational equivalence conditionobtained in the full-information RE model (see Backus, Routledge, and Zin 2004).That is, under the assumption that the agent distrusts the Kalman filter equation, theOE result obtained under full-information RE still holds under RI.27

HST (1999) show that as far as the quantity observations on consumption andsavings are concerned, the robustness version (ϑ > 0 or α > 0, β̃ ) of the PIH modelis observationally equivalent to the standard version (ϑ = ∞ or α = 0,β = 1/R)of the PIH model for a unique pair of discount factors.28 The intuition is that in-troducing a preference for risk-sensitivity (RS) or a concern about robustness (RB)increases savings in the same way as increasing the discount factor, so that the dis-count factor can be changed to offset the effect of a change in RS or RB on consump-tion and investment.29 Alternatively, holding all parameters constant except the pair(α,β ), the RI version of the PIH model with RB consumers (ϑ > 0 and βR = 1)is observationally equivalent to the standard RI version of the model (ϑ = ∞ andβ̃ > 1/R).

Proposition 3. Let

β̃ =1R

1−Rω2η/(2ϑ)1−R2ω2η/(2ϑ)

=1R

1−Rαω2η1−R2αω2η

>1R.

Then consumption and savings are identical in the RI, RB-RI, and RS-RI mod-els.

5 Conclusions

In this paper we show that a state-space representation can be explicitly derived froma classic macroeconomic framework which has incorporated model uncertainty dueto concerns for model misspecification (robustness or RB) and state uncertainty dueto limited information constraints (rational inattention or RI). We show the state-space representation can also be used to quantify the key model parameters. Severalapplications are also provided to show how this general framework can be used toaddress a range of interesting economic questions.

27 Note that the OE becomesαθ

1− (1−θ)R2=

12ϑ

,

if we assume that the agents distrust the income process hitting the budget constraint, but trust theRI-induced noise hitting the Kalman filtering equation.28 HST (1999) derive the observational equivalence result by fixing all parameters, including R,except for the pair (α,β ).29 As shown in HST (1999), the two models have different implications for asset prices becausecontinuation valuations would alter as one alters (α,β ) within the observationally-equivalent setof parameters.


Acknowledgements This paper is prepared for the book volume “State Space Models — Appli-cation in Economics and Finance” in a new Springer Series. Luo thanks the Hong Kong GRF undergrant No. 748209 and 749510 and HKU seed funding program for basic research for financial sup-port. All errors are the responsibility of the authors. The views expressed here are the opinions ofthe authors only and do not necessarily represent those of the Federal Reserve Bank of Kansas Cityor the Federal Reserve System.

Appendix

6 Solving the Current Account Model Explicitly under ModelUncertainty

To solve the Bellman equation (7), we conjecture that

v(st) =−As2t −Bst −C,

where A, B, and C are undetermined coefficients. Substituting this guessedvalue function into the Bellman equation gives

−As2t −Bst −C = maxctmin

νt

{−1

2(c− ct)2 +βEt

[ϑν2t −As2t+1−Bst+1−C

]}.

(48)We can do the min and max operations in any order, so we choose to do the mini-mization first. The first-order condition for νt is

2ϑνt −2AEt[ωζ νt +Rst − ct

]ωζ −Bωζ = 0,

which means that

νt =B+2A(Rst − ct)

2(

ϑ −Aω2ζ) ωζ . (49)

Substituting (49) back into (48) gives

−As2t −Bst−C =maxct

−12 (c− ct)2 +βEtϑB+2A(Rst − ct)

2(

ϑ −Aω2ζ) ωζ

2−As2t+1−Bst+1−C ,

wherest+1 = Rst − ct +ζt+1 +ωζ νt .

The first-order condition for ct is

(c− ct)−2βϑAωζ

ϑ −Aω2ζνt +2βA

(1+

Aω2ζϑ −Aω2ζ

)(Rst − ct +ωζ νt

)+βB

(1+

Aω2ζϑ −Aω2ζ

)= 0.


Using the solution for νt the solution for consumption is

ct =2AβR

1−Aω2ζ/ϑ +2βAst +

c(

1−Aω2ζ/ϑ)+βB

1−Aω2ζ/ϑ +2βA. (50)

Substituting the above expressions into the Bellman equation gives

−As2t −Bst −C

=−12

(2AβR

1−Aω2ζ/ϑ +2βAst +

−2βAc+βB1−Aω2ζ/ϑ +2βA

)2

+βϑω2ζ(

2(

ϑ −Aω2ζ))2

2AR(

1−Aω2ζ/ϑ)

1−Aω2ζ/ϑ +2βAst +B−

2c(

1−Aω2ζ/ϑ)

A+2βAB

1−Aω2ζ/ϑ +2βA

2

−βA

R

1−Aω2ζ/ϑ +2βAst −

−Bω2ζ/ϑ +2c+2Bβ

2(

1−Aω2ζ/ϑ +2βA)2 +ω2ζ

−βB

R1−Aω2ζ/ϑ +2βA

st −−Bω2ζ/ϑ +2c+2Bβ

2(

1−Aω2ζ/ϑ +2βA)−βC.

Given βR = 1, collecting and matching terms, the constant coefficients turn out tobe

A =R(R−1)

2−Rω2ζ/ϑ, (51)

B =− Rc1−Rω2ζ/(2ϑ)

, (52)

C =R

2(

1−Rω2ζ/2ϑ)ω2ζ + R

2(

1−Rω2ζ/2ϑ)(R−1)

c2. (53)

Substituting (51) and (52) into (50) yields the consumption function. Substituting(53) into the current account identity and using the expression for st yields the ex-pression for the current account.

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work,” Macroeconomic Dynamics 12 (supplement 1), 126-135.14. Leitemo, Kai, Ulf Soderstrom (2008), “Robust Monetary Policy in a Small Open Economy,”

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Table 1 Emerging vs. Developed Countries (Averages)

A: Emerging vs. Developed Countries (HP Filter)σ(y)/µ(y) 4.09(0.23) 1.98(0.09)σ(∆y)/µ(y) 4.28(0.23) 1.89(0.07)ρ(yt ,yt−1) 0.53(0.03) 0.66(0.02)ρ(∆yt ,∆yt−1) 0.28(0.05) 0.46(0.03)σ(c)/σ(y) 0.74(0.02) 0.59(0.02)σ(∆c)/σ(∆y) 0.71(0.02) 0.59(0.02)σ(ca)/σ(y) 0.79(0.03) 0.85(0.04)ρ(c,y) 0.85(0.02) 0.78(0.02)ρ(cat ,cat−1) 0.30(0.05) 0.41(0.03)ρ(ca,y) −0.59(0.05) −0.35(0.04)ρ(

cay ,y)

−0.54(0.04) −0.36(0.04)

B: Emerging vs. Developed Countries (Linear Filter)σ(y)/µ(y) 7.97(0.40) 4.79(0.22)σ(∆y)/µ(y) 4.28(0.23) 1.89(0.07)ρ(yt ,yt−1) 0.79(0.02) 0.89(0.01)ρ(∆yt ,∆yt−1) 0.28(0.05) 0.46(0.03)σ(c)/σ(y) 0.72(0.02) 0.58(0.02)σ(∆c)/σ(∆y) 0.71(0.02) 0.59(0.02)σ(ca)/σ(y) 0.54(0.03) 0.65(0.04)ρ(c,y) 0.88(0.02) 0.85(0.02)ρ(cat ,cat−1) 0.53(0.04) 0.71(0.02)ρ(ca,y) −0.17(0.06) −0.08(0.05)ρ(

cay ,y)

−0.32(0.05) −0.20(0.04)


Table 2 Implications of Different Models (Emerging Countries)

Data RE RB RB+RI RB+RI RB+RI RB+RI(θ = 0.9) (θ = 0.8) (θ = 0.7) (θ = 0.5)

(λ = 1)ρ(ca,y) 0.13 1.00 0.62 0.57 0.56 0.56 0.58ρ(cat ,cat−1) 0.53 0.80 0.74 0.57 0.50 0.45 0.36σ(ca)/σ(y) 0.80 0.71 0.49 0.52 0.55 0.59 0.79σ(∆c)/σ(∆y) 1.35 0.28 0.90 0.89 0.89 0.91 1.36

(λ = 0.5)ρ(ca,y) 0.13 1.00 0.62 0.59 0.58 0.59 0.64ρ(cat ,cat−1) 0.53 0.80 0.74 0.63 0.59 0.55 0.46σ(ca)/σ(y) 0.80 0.71 0.49 0.50 0.52 0.53 0.64σ(∆c)/σ(∆y) 1.35 0.28 0.90 0.85 0.81 0.79 0.99

(λ = 0.1)ρ(ca,y) 0.13 1.00 0.62 0.61 0.60 0.61 0.67ρ(cat ,cat−1) 0.53 0.80 0.74 0.67 0.64 0.62 0.56σ(ca)/σ(y) 0.80 0.71 0.49 0.49 0.50 0.51 0.57σ(∆c)/σ(∆y) 1.35 0.28 0.90 0.84 0.79 0.75 0.82

Table 3 Theoretical corr(c,c∗) from Different Models

Data RE RB RB+SU RB+SU RB+SU(θ = 0.9) (θ = 0.6) (θ = 0.3)

Canada(λ = 1) 0.38 0.41 0.33 0.27 0.17 0.12(λ = 0.5) 0.38 0.41 0.33 0.31 0.26 0.23(λ = 0.1) 0.38 0.41 0.33 0.32 0.32 0.32Italy(λ = 1) 0.25 0.54 0.50 0.42 0.27 0.19(λ = 0.5) 0.25 0.54 0.50 0.48 0.41 0.36(λ = 0.1) 0.25 0.54 0.50 0.50 0.50 0.49UK(λ = 1) 0.21 0.69 0.45 0.38 0.25 0.17(λ = 0.5) 0.21 0.69 0.45 0.44 0.38 0.32(λ = 0.1) 0.21 0.69 0.45 0.46 0.46 0.45France(λ = 1) 0.46 0.51 0.49 0.40 0.26 0.18(λ = 0.5) 0.46 0.51 0.49 0.46 0.40 0.34(λ = 0.1) 0.46 0.51 0.49 0.49 0.48 0.48Germany(λ = 1) 0.04 0.45 0.40 0.33 0.22 0.15(λ = 0.5) 0.04 0.45 0.40 0.38 0.33 0.29(λ = 0.1) 0.04 0.45 0.40 0.40 0.40 0.40

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